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DOI: 10.4230/LIPIcs.STACS.2023.53

Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

Authors: Yaniv Sadeh and Haim Kaplan

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is 4/3. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of 2 on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be Ω(m ⋅ log log n) where n is the number of items in the tree and m is the number of queries.

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Yaniv Sadeh and Haim Kaplan. Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{sadeh_et_al:LIPIcs.STACS.2023.53,
  author =	{Sadeh, Yaniv and Kaplan, Haim},
  title =	{{Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{53:1--53:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.53},
  URN =		{urn:nbn:de:0030-drops-177055},
  doi =		{10.4230/LIPIcs.STACS.2023.53},
  annote =	{Keywords: Binary Search Trees, Greedy Future, Geometric Greedy, Lower Bounds, Dynamic Optimality Conjecture}
}

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Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

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